Vol. 33, No. 1, 1970

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On secondary characteristic classes in cobordism theory

Wei-lung Ting

Vol. 33 (1970), No. 1, 249–253

This paper introduces into cobordism theory a new notion borrowed from ordinary cohomology theory. Specifically, let ξ be a U(n)-bundle over the CW-complex X. Let E and E0 be the total spaces of the associated bundles whose fibers are respectively the unit disc E2n Cn and the unit sphere S2n1 Cn. The classifying map for ξ gives rise to an element Uξ ΩU2n(E,E0). One defines the Thom isomorphism φ : ΩUq(X) ΩUq+2n(E,E0) by φ(x) = (px)Uξ and Euler class, e(ξ) of ξ, by e(ξ) = p∗−1j(Uξ). For each α = (α12,), let ofα(ξ) ΩU2|α|(X) be the Conner-Floyd Chern class of ξ, and Sα : ΩUq(X,Y ) ΩUq+2|α|(X,Y ) be the operation defined by Novikov. Then one has the relation, Sα(e(ξ)) = cfα(ξ) e(ξ). Now if ξ is a bundle such that e(ξ) = 0, then one can define a secondary characteristic class

Σα(ξ) ∈ Ω∗U (X ) mod (Sα − cfα (ξ))Ω∗U(X )

by using the above relation. The object of this paper is to study some of the properties of such secondary characteristic classes.

Mathematical Subject Classification
Primary: 57.32
Received: 29 August 1969
Published: 1 April 1970
Wei-lung Ting